Lemma 2.23 [1]}
For the \(H\) -action on \(Y\)  with respect to an ample adapted linearisation \(\mathcal {L}\) , we have for \(z \in Z_{\min }\)  and \(u\in U\)  that \(uz \in Z_{\min }\)  if and only if \(u\in \operatorname{Stab}_U(z)\) . Furthermore,
\(\operatorname{Stab}_H(z) = \operatorname{Stab}_U(z)\rtimes \operatorname{Stab}_L(z)\) 
